F\\Q = F \F(Z)\\K(Z, Z)\-Q/2Dm(Z)

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 3, March 1980

ON BOUNDEDNESS OF INTEGRABLE AUTOMORPHIC FORMS IN C

SU-SHING CHEN

Abstract. We give a necessary and sufficient condition for an integrable automorphic form on a bounded symmetric domain D in C to be bounded.

The question on the boundedness of integrable automorphic forms on the unit disk in C1 has been investigated in [1], [5], [6], [7], [8], [9] and many other papers. Integrable automorphic forms on the bounded homogeneous domains in C" have been considered by Earle [2] and Selberg [10]. In this paper, we shall study the boundedness of integrable automorphic forms on the bounded symmetric domains inC". Let D be a bounded symmetric domain in C" with Bergmen kernel function k(z, w), where z and w represent «-tuples (z,, . . . , zn) and (wx, . . . , wn) respec- tively. For every holomorphic automorphism g of Aut(Z)), we have k(z, w) = k(gz,gw)g'(z)g'(w), where g'(z) is the complex Jacobian of the automorphism g. The volume element dm(z) = k(z, z)dz is invariant under the group Aut(Z)) of all holomorphic automorphisms of D, where dz is the euclidean volume element of D. Let r be a discrete subgroup of Aut(£>). We choose a fundamental domain R for T so that 32? n D has zero volume. A function/holomorphic on D is said to be an automorphic form of dimension -2q if f(yz)y'(z)q = f(z) for all z in D and y in T. We denote by A (T) the space of integrable forms, i.e., the set of all holomorphic automorphic forms/of dimension -2q such that

\\f\\q = f \f(z)\\k(z, z)\-q/2dm(z) < *o.

We denote by B (T) the space of bounded forms, i.e., the set of all forms of dimension -2q such that

H/IL- sup|/(z)/;(z,z)-«/2|. ZED We refer to the paper [2] of Earle for notations and basic facts. In particular, q is any integer > 2 so that all formulas in [2] are valid. c(q) is a certain constant depending only on q. The following theorem is a generalization of a result of Metzger and Rao [7].

Received by the editors January 12, 1979. AMS (MOS) subject classifications (1970). Primary 32N15. Key words and phrases. Automorphic form, bounded symmetric domain. © 1980 American Mathematical Society 0002-9939/80/0000-0111/$01.75 342

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INTEGRABLE AUTOMORPHIC FORMS 343

Theorem. Let D be a bounded symmetric domain in C" and let V be a discrete subgroup of Aut(D). For q > 2, Aq(T) c Bq(T) if and only if sup \k(z, z)~qa(z, z)\ < oo, (**) zeD where a(z, w) = c(q)'ZyETk(yz, w)qy'(z)q. Proof. According to Theorem 3.1, Corollary 5.2 and Theorem 7 of [2], the function a(z, w), for a fixed w in D, belongs to Aq(T) and Bq(T). If/is in Aq(T), then by [2], we have

/(") = c(q) [ f(z)k(w, z)"k(z, z)-« dm(z) JD

= c{

= c(q) f 2 f(yz)k(w, yz)qk(z,z)~qy'(z)qY(z) " dm(z) JR 7er

= c(q)[ 2 k(w,yz)q7(z)qf(z)k(z,z)-qdm(z) JR yer

= c(q) Í f(z) a(z, w) k(z, z)~q dm(z). JR Consequently,

|/(w)| \k(w, w)'q/2\ < c(q)\\f\\q sup \a(z, w)\ \k(w, w)\~"/2\k(z, z)\~ql2. z.weD Since a(z, w) is in Aq(T) for a fixed w in D,

a(w, w) = c(q) I \a(z, w)\2k(z, z)~q dm(z) J R and

a(z, w) = c(q) I a(w', w) a(w', z) k(w', w')~q dm(w'). JR

The Schwarz inequality implies that \a(z, w)\2 < \a(z, z)a(w, w)\. Thus / is in Bq(T). Conversely, by the closed graph theorem, we have a bounded linear map from Aq(T) into Bq(T). Thus, there exists a positive constant C such that for all / in Aq(T) and z in D \f(z)\\k(z,z)-q/2\

\a(z,w)\\k(z,z)-q/2\

\a(w, w)\ \k(w, w)-q/2\

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 344 SU-SHING CHEN

But

ll«(-, *0||, < c(q) [ S \k(yz, w)\q\y'(z)\q\k(z, z)~i/2| dm(z) JR Yer

= c(?)rirc(z,»vriirv(z,z)-?/2i^(z) JD

= c{q)/c(q/2)\k(w, w)q/2\ by a formula in [2]. Thus (**) is satisfied.

References 1. C J. Earle, A reproducing formula for integrable automorphic forms, Amer. J. Math 88 (1966), 867-870. 2._, Some remarks on Poincaré series, Compositio Math. 21 (1969), 167-176. 3. R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, EnglewoodCliffs, N. J., 1965. 4. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. 5. J. Lehner, On the Aq(T) c Bq(T) conjecture for infinitely generated groups, Discontinuous Groups and Riemann Surfaces (Proc. Conf. Univ. Maryland, College Park, Md., 1973), Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N. J., 1974, pp. 283-288. 6. _, Automorphic forms, Discrete Groups and Automorphic Functions, Academic Press, New York, 1977,pp. 73-120. 7. T. A. Metzger and K. V. Rajeswara Rao, On integrable and bounded automorphic forms, Proc. Amer. Math. Soc. 28 (1971),562-566. 8. D. Neibur and M. Sheingorn, Characterization of Fuchsian groups whose integrable forms are bounded,Ann. of Math. (2) 106 (1977), 239-258. 9. C. Pommerenke, On inclusion relations for spaces of automorphic forms, Advances in Complex Function Theory, Lecture Notes in Math., No. 505, Springer-Verlag, Berlin and New York, pp. 92-100. 10. A. Selberg, Automorphic functions and integral operators, Seminar on Analytic Functions, Institute for Advanced Study, Princeton, N. J., 1957, pp. 152-161.

Department of Mathematics, University of Florida, Gainesville, Florida 32611

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